If a predictor has high tolerance, that means there’s a strong unique effect of that predictor, right?

No. The test of the regression coefficient is always the test of the predictor’s unique contribution, whether or not there’s multicollinearity. If you have a significant effect for a predictor with low tolerance, that test still indicates what it always does – that the predictor’s unique relationship (taking into account everything else in the model) with the criterion is significantly different from zero. A high tolerance suggests that there is minimal overlap between predictors, so the test of that predictor’s unique effect (the regression coeff) is similar to a test of the bivariate relationship between that predictor and the criterion. That makes the estimate of the effect more stable (changing the model doesn’t change the effect as much, since it’s reflecting the bivariate relationship between the predictor and criterion, which doesn’t depend on the model), and decreases the SE. So having a high tolerance makes it easier to test the unique effect of a predictor, since it’s not being masked by overlap with other variables, but a high tolerance doesn’t actually indicate anything about the strength (or significance) of the relationship between a predictor and the criterion.